"History, perhaps?  In any case, the sign of a NaN isn't useful information in the same way that the sign of an infinity is.  The IEEE 754 standard explicitly refuses to attach any meaning to the sign bit of a NaN.  And if we were aiming for a full and faithful representation of NaNs, we'd want to output the payload, too (which is just about as meaningless / meaningful as the sign bit)."

So I understand that adding a math.neg_nan would be useless. As adding one constant per possible "NaN" value :-) If I recall correctly the IEEE 754 standard, there is not single NaN value, but a range of NaN.

"Two kinds of NaN: a quiet NaN (qNaN) and a signaling NaN (sNaN). A NaN may carry a payload that is intended for diagnostic information indicating the source of the NaN. The sign of a NaN has no meaning, but it may be predictable in some circumstances." says Wikipedia.

Well, the current definition of math.nan makes sense, it's the same value than float("nan").

Note: On python-ideas, I asked if math.nan and math.inf should be singleton (as it was requested for float("0.0") in issue #4024). The answer is no.